Jean-Noel BACRO (Montpellier), An estimator for the extremal coefficient function of stationary random fields
Spatial data are now widely used in practice and spatial processes are often used in applications such as hydrology, climatology or more generally environmental studies. Usually, the characterization of the behaviour of the related extremes is fundamental. Well known dependence measures for bivariate extremes can be generelized to the spatial case and some estimators of the extremal coefficient at distance h have been recently proposed (Schlather and Tawn (2003), Poncet, Cooley and Naveau (2005)). We propose a new estimator for the extremal coefficient function. Statistical properties are derived and illustrated through simulation studies. This work comes from a collaboration with C. Lantuéjoul (Ecole des Mines, Paris) and L. Bel (Orsay university).

Jan BEIRLANT (Louvain), On the adaptive choice of thresholds in extremes and bandwidths in density estimation
In recent extreme value literature several methods can be found for choosing the number of extremes or thresholds adaptively in extreme value estimation techniques. Some of them are based on goodness-of-fit test statistics. We review the known methods and indicate links with testing when available. Almost all proposals are restricted to the univariate case. We also briefly discuss the multivariate case. Correspondingly, in density estimation, the problem of choosing a bandwidth or the number of nearest neighbors addresses a similar problem. We explain the correspondance between both settings and indicate what these two areas can offer each other.

Margarida BRITO (Porto), Uniform estimation of isobars
The ordering of a multivariate sample is not natural and several definitions have been proposed in the literature. The extension of the definition of univariate quantiles to multivariate distributions is, in this way, not straightforward and several approaches have been used. Here we consider a multivariate sample ordered according to an increasing family of conditional quantile surfaces, called isobar surfaces, and we shall discuss the uniform a.s. consistency of simple nonparametric estimators of these isobar surfaces. In particular, we will see that, under some regularity conditions, the strong behaviour of these isobar estimators is determined by the corresponding behaviour of suitable defined histogram-type estimators of the underlying conditional distribution function. Consequently, we begin our study by investigating the weak and strong limiting behaviour of the conditional distribution function estimators. (with Marie-Françoise Barme, Lille1)

Anne-Laure FOUGERES (Nanterre), Mixtures models for extremes
This work develops ``logistic'' multivariate extreme value distributions obtained by mixing with positive stable distributions. The mixing variables are used as a modelling tool, to introduce new classes of mixture models for extreme value data, and as a road to better understanding and use of the models. A distinguishing feature is that the models lead to extreme value distributions both conditionally on the mixing variables and unconditionally. In many situations this is important for easy interpretation and extrapolation and it is natural for data obtained as maxima of other underlying variables. We develop analogues of the components of variance models in ANOVA, and new time series, spatial, and continuous parameter models for extreme value data. The results are applied to data from a pitting corrosion investigation. Joint work with J. NOLAN (American University, Washington DC, USA) and H. ROOTZEN (Chalmers University, Gothenburg, Sweden).

Isabel FRAGA-ALVES (Lisbonne), Testing Max-Domains of Attraction
Here we deal with the semi-parametric approach to the problem of statistical choice of extreme domains of attraction. We revisit the two-sided problem of testing Gumbel domain against Fréchet or Weibull domains. Relying on concepts of regular variation theory, we present some asymptotic properties of Hasofer and Wang's test statistic based on the k- upper extremes taken from a sample of size n, when k behaves as an intermediate sequence k=kn rather that remaining fixed while the sample size increases. In the process, a Greenwood type test statistics is proposed which turns out to be useful in discriminating heavy-tailed distributions. Next to this, we introduce a complementary test statistic which is merely a ratio of the maximum to the mean of the sample of the excesses above some random threshold. The finite sample behaviour of the three referred testing procedures is evaluated on the basis of a simulation study. Illustration is carried out with three real published data sets.

Stéphane GIRARD (Grenoble), Statistical inference for Weibull tail-distributions
We focus on a particular class of distributions: the Weibull Tail-distributions whose cumulative hazard function is regularly varying. This family includes for instance Weibull, normal, and gamma distributions. These dis< tributions are indexed by a parameter called the Weibull tail coefficient. We propose first some new estimators of this coefficient and we establish their asymptotic properties. Second, we introduce an exponential regression model allowing to reduce the bias of the above estimators. Finally, we address the problem of estimating extreme quantiles of these distributions. Some simulations are also provided to illustrate the behavior of these estimators. (joint work with Jean Diebolt, Laurent Gardes and Armelle Guillou)

Ivette GOMES (Lisbonne), A reduced bias' tail index and associated quantile estimator
We shall deal with bias reduction techniques for heavy tails, trying to improve the performance of classical high quantile estimators. High quantiles depend strongly on the tail index, and the classical estimators of the tail index usually exhibit the same type of pattern : high variance for high thresholds, i.e. for small values of k, the number of top order statistics used for the estimation, high bias for large k and consequently, very sharp mean squared errors. Recently, new interesting classes of reduced bias' estimators have been introduced in the literature. In those classes, the second order parameters in the bias are estimated at a level k' of a larger order than that of the level k at which we compute the tail index estimators, and doing this, it is possible to keep the asymptotic variance of the new estimators equal to the asymptotic variance of the Hill estimator, the maximum likehood estimator of the parameter, under a strict Pareto model. We introduce here a similar class of estimators and this enables us again to build new classes of high quantiles' estimators. The asymptotic distributional properties of the proposed estimators are derived and the estimators are compared with alternative ones, not only asymptotically, but also for finite samples through Monte Carlo techniques. We also provide an illustration of the behaviour of these estimators for different sets of real data.

Rolf-Dieter REISS (Siegen), Multivariate Generalized Pareto Distributions
The statistical modeling by univariate generalized Pareto distributions, due to J. Pickands, was one of the most fruitful innovations in extreme value theory during last decades. Less is known in the multivariate framework. We deduce the multivariate versions of generalized Pareto distributions from limiting exceedance processes and discuss some properties in conjunction with a new spectral decomposition of multivariate distributions.

François G Schmitt (Lille1), Extremes and turbulences ; stretched exponentials or hyperbolic laws?
In turbulence, an adequate modelling of extremes is important in several fields: meteorology, wind energy, aeronautics, etc. The turbulent wind field is very intermittent, and has a correlated dynamics on a large range of scales; its extremes are modelled differently according to the community. In the field of wind energy, the wind field itself is studied, and its extremes are usually considered as close to a Weibull distribution. In the field of fully developed turbulence, it is rather the increments of the velocity field that are considered. When taking increments, the probability of wind fluctuations have a tail that decreases much more slowly. There are different schools: some authors model these extremes using stretched exponentials, while others advocate rather hyperbolic laws (also called Frechet or Pareto laws). After a short overview of the results published in different communities, I will analyze turbulent databases from marine and atmospheric measurements. I will show that hyperbolic laws seem the closest to the data for turbulent velocity increments.